Coordinate conversion from the RAH coordinate system to the HGI one, as it
has been done for non-geocentric spacecraft from the TASPD SINP MSU data
collection, is described below.
RAH coordinate system is a heliocentric coordinate system in which
coordinates of a spacecraft are defined by:
R - radial distance between sun and spacecraft (in km or millions km);
A - ecliptic projection of the angle Earth-Sun-spacecraft (in degrees);
H - height above/under ecliptic plane (in km or millions km).
There is also a RAT coordinate system which differs from RAH only by that
T - an angle spacecraft-Sun-
in degrees - is used instead of H.
HGI coordinate system, as follows from
http://nssdc.gsfc.nasa.gov/cohoweb/html/cw_data.html#general,
is: "
Heliographic Inertial Coordinate System (HGI):
The HGI coordinates are Sun-centered and inertially fixed with respect
to an X-axis directed along the intersection line of the ecliptic
and solar equatorial planes. The solar equator plane is inclined
at 7.25 degrees from the ecliptic. This direction was towards
ecliptic longitude of 74.367 degrees on 1 January 1900 at 1200 UT;
because of precession of the celestial equator, this longitude
increases by 1.4 degrees/century. The Z axis is directed
perpendicular and northward from the solar equator, and the Y-axis
completes the right-handed set. This system differs from the usual
heliographic coordinates (e.g. Carrington longitudes) which are
fixed in the frame of the rotating Sun.
"
In the coordinate conversion, the following assumptions have been used:
- the Earth goes up its one-year orbit for 365.25 days;
- vernal equinox is passed by the Earth on 1981/03/20 at 17:03.1 UT;
- the line of intersection of the ecliptic and solar equatorial planes
has a longitude of 75.5 degrees (from the direction to vernal equinox);
- the solar equator plane is inclined at 7.25 degrees from the ecliptic.
The conversion is being performed in two steps. First is a transfer to
the XYZ axes of the ecliptic coordinate system (X - along the direction to
vernal equinox; Z - towards ecliptic's North pole; Y - is right-hand
complementation). Second step is a conversion from the ecliptic's to
the HGI system.
On the first step, the following procedure is used:
W=360.0/(365.25*60*60*24)*pi/180.0; # Earth's angular velocity [rad/s]
E=0.0+W*(UTc-UTb); # Earth's current angular position [rad] in ecliptic system
P=E+A*pi/180.0; # spacecraft's current angular position [rad] in ecliptic system
X=R*cos(P); # current X-coordinate of spacecraft in ecliptic system
Y=R*sin(P); # current Y-coordinate of spacecraft in ecliptic system
Z=H; # current Z-coordinate of spacecraft in ecliptic system
Here pi is pi; R,A,H - as defined in RAH system; UTc [s] is current time (UT)
relative to the basic time UTb [s] (on 1981/03/20, 17:03.1 UT).
On the second step, the following procedure is used:
p=75.5*pi/180.0;
t=7.25*pi/180.0;
x= X*cos(p) +Y*sin(p);
y=-X*sin(p)*cos(t)+Y*cos(p)*cos(t)+Z*sin(t);
z= X*sin(p)*sin(t)-Y*cos(p)*sin(t)+Z*cos(t);
Here pi is pi; X,Y,Z are taken from the previous step; x,y,z - HGI coordinates.
In practice, the RAH/RAT coordinates, digitized from low quality in-paper
plots, gave rather unsmooth curves. To make the data more realistic, we
have performed additional correction of the digitized trajectory data.
The correction was rather arbitrary; several points have been excluded in
each data set for the sake of the remained rarely placed points give a more
smooth resulting curve.
Finally used coordinates in RAH/RAT system are presented in the files
id_coord.trj in subdirectories, together with derived from them coordinates
in HGI system. The used legends are explained in the file sc_coordinates.txt.
After conversion, the HGI coordinates were interpolated to the 4-hour time
grid and then a data synchronization procedure has been performed.
A bi-parabolic method is used for interpolation: four points i1, i2, i3 and i4
are involved on each step; a line between the points i2 and i3 is derived as
a mean of two parabolic curves based on the sets of points (i1,i2,i3) and
(i2,i3,i4) respectively; the linear weight functions w1 and w2 are used for
computing the mean values; w1 is applied to the 1st parabola and w2 - to
the 2nd parabola; w1=1 at the point i2 and w1=0 at the point i3; w2=0 at
the point i2 and w2=1 at the point i3.
Synchronization is based on the averaging in a moving window. Bell-like weight
function (based on cosine function) is used in this procedure. A one-day width
of window is applied for 4-hour gridded coordinates taken from interpolation.